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The case is heard under the Supreme Court's appellate jurisdiction.
Most US Supreme Court cases fall under its appellate jurisdiction, meaning the case started in a trial court (U.S. District Court or an appropriate state court), and has exhausted its lower court appeals. Under these circumstances, the case is "On appeal to the Supreme Court of the United States."
The Supreme Court only accepts "federal question" cases, meaning the appeal must involve one or more (usually unresolved) issues of federal or constitutional law.
If a case is first heard by the US Supreme Court, it is heard under the Court's original jurisdiction.
When a case is heard for a 2nd time in supreme court, its called an appeal.
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Appellate jurisdiction
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Writ of Certiorari
The written explanation for the US Supreme Court's decision is called an opinion.See Related Questions for an expanded explanation.
The Supreme Court is the court of last resort. When all appeals and lower courts have heard and ruled on a case it may go to the Supreme Court, but the court doesn’t have to hear it and may let the lower ruling stand or kick it back to the lower federal court.
The Supreme Court
The US Supreme Court has heard more than 30,000 cases since its inception in 1789 (no cases were heard for the first few years).
The party must request that the court grant a writ of certiorari.